task quarterly99

TASK Quarterly 3 No 3 (1999), 343-360

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APPLICATION OF THE VORTEX-IN-CELL METHOD FOR THE SIMULATION OF TWO-DIMENSIONAL VISCOUS FLOW

Henryk Kudela,

Wrocław University of Technology,Wybrze�e Wyspia�skiego 27,50-370 Wrocław, Poland

ABSTRACT.In the paper the vortex in cell method for the simulation of the viscous flow in a complex geometry was described. Vorticity field is approximated by the collection of the particles that carries the circulation. The local velocity of a particle was obtained by the solution of the Poisson equation for the stream function by the grid method and then interpolation of velocity from the grid nodes to the vortex particle position. The Poisson equation for the stream function was solved by fast elliptic solvers. To be able to solve the Poisson equation in a region with a complex geometry, the capacitance matrix technique was used. The viscosity of the fluid was taken in a stochastic manner. A suitable stochastic differential equation was solved by the Huen method. The non-slip condition on the wall was realized by the generation of the vorticity. The program was tested by solving several flows in the channels with a different geometry and at a different Reynolds number. Here we present the testing results concerning the flow in a channel with sudden symmetric expansion, for the flow in channel with backward step, and the flow over building systems.

1. INTRODUCTION

The vortex method belongs to the particle methods. It means that for the solution of the equation of motion we utilized the „particles“ that are called vortex, which serve as a carrier of circulation [5, 6, 9, 15, 19, 25]. Calculations are carried out in lagrangian coordinates. Generally the vortex methods are divided into the direct method in which the velocity of the vortex particle are calculated by the summation of the contribution from all particles that exist at the flow by virtue of the Biot-Savart law [7] and the method called vortex-in-cell method [9]. Due to the fact that the number of operations in direct vortex method are in each time step proportional to the square of the number of particles that are in the flow �0(N2), computational time for solutions of the specific problems are very large. On the other hand, in the vortex in cell method [9], the velocity is calculated through differentiation of the stream function, which is obtained by the solution of the Poisson equation on the numerical grid. The number of operations per one time step are proportional to O(N+MlogM), where M being the number of points on the mesh, and that results in essentially reducing computational time. Here we described the vortex-in-cell method. We wrote the general proposed program on the basis of the vortex in cell method and tested it by solution of several problems and compared the results with experimental data or with numerical results obtained by a different method.

2. DESCRIPTION OF THE VORTEX IN CELL METHOD

The non-dimensional equations of incompressible fluid motion in two-dimensional space

transformed to the vorticity transport equations [7] take the form:

)1( 1

= )u ( ωωω ∆∇⋅∂∂

Re

+ t


2

where � is the non-zero component of the vorticity vector, u = (u,v) is the velocity divided by the uniform inlet velocity U, � is the stream function, t - is the time and Re is the Reynolds number defined

as Re=Uh/v where v is the coefficient of kinematic viscosity. The vortex method is based on the so-called viscous splitting algorithm. First, the Euler equation

(�=0) is solved; then the diffusion equation is solved. Due to fact that the diffusion is taken into account in the stochastic manner, we can interpret the equation of fluid motion in terms of the stochastic processes. One can note that due to the incompressibility of the fluid,��u=0, the equation (1) can be rewritten as:

Equation (3) is identical, with respect to form, to the forward Kolmogorow -Fokker-Planck equation that describes the probability density called transition density for the stochastic (Marcov) process:[17]:

where P is probability, G(x,t;�,0) is a solution of equation (3). In the theory of the stochastic differential equation, it is shown that the stochastic process that is described by a stochastic differential equation (in the Ito sense) [17]

has the transition density function that satisfies equation (3), where W means the Wiener process. So equation (5) described the convection and diffusion process and can be regarded as the fluid motion equation. For the solution (5) we used a viscous splitting algorithm [6, 19]: velocity (drift) is calculated for inviscid flow and the last term is added in order to take into account the diffusive property of the viscosity of the fluid. In the present work instead of the commonly used Euler scheme that has only order of convergence 0.5, for the solution of equation (5) we use the generalized Huen scheme that has the order one [17]:

where �Wn is an increment of the Wiener process, and �t is a time step. It is well known that the increments of the Wiener process are the independent Gaussian random variables with mean E(�Wn )=0 and variance E((�Wn)2)=�t; so it is relatively easy to generate it by pseudo-random generator of numbers with uniform distribution and using the Box-Muller transformation [17]. Now we describe the VIC algorithm for obtaining the inviscid velocity field u(u,v). Vorticity �(x,y) is approximated by the linear combination of the Dirac measures:

)3( Re1

= )u ( ωωω ∆⋅∇∂∂

+ t

)2(x

- = v ,

y

=u , - = ∂∂

∂∂∆ ψψωψ

)4(x)0,())(( d t; ,xG = t ,A t XPA

α�∈

(5)2

+ ) ,x u()X( dWRe

dt t = td

( )

WtRe

t

WtRe

t

npn

pp

npn

pnn

pnp

2

+ ) x ( u x xwhere

)6( 2

)x( u ) (x u 21

x x

*

* 1

∆∆∆+=

∆∆+∆++=+

)7()x(),x(x(x) 2p

np

np

pp

n h ωδω =Γ−Γ= �


3

where p is the number of the vortex particle. Approximation (7) is understood in the sense of measure on R2 [25]:

We assumed that we were able to solve the Poisson equation for the stream function (2) by the finite difference method. The computation goes as follows: 1) At first the redistribution of the mass of vortex particles on the grid nodes is done:

where �j(x)=�((x-xj)/h), is a B-spline of order m [26, 18]. For m = 1 the B-spline has the form: and the redistribution process (9) corresponds to a well known area-weighted interpolation scheme [8]. In the present work we use the B-spline of the first order in the cells that adjoin the boundary of the

domain flow and outside these cells we used the B-spline of the 3rd order that takes the form [18, 26]:

The B-splines satisfy: �(x)=�(-x) and ��(x)dx=1. To obtain the vorticity in the grid node, we should divide the circulation of the node obtained from (9) by the volume of the cell h2. Instead of that, Cottet [10] proposed, in order to overcome some difficulties related to the accumulation of the particles, the calculation of the volume of the node through position of the particles around the node:

Then the vorticity in j node is calculated as:

)8()( )( 2

p

hxdxx p�� ≈ ωω

)9()(x np

pjp

nj � Γ=Γ ϕ

)10(1011

)(�� ≤

> | x | for

| x | for | x | - = x ϕ

)12()h

x - x ( h = ) x ( h = Jpj2

ppj

2

pj ϕϕ ��

)13() x ( h

x =

J =

pj2

p

pjpp

j

jj

) (

ϕ

ϕω

�

�ΓΓ

)11(

2||0

2||134

||2||61

1||,32

||21

)( 23

23

��

�

��

�

�

≥

<<+−+−

≤+−

=

x

xxxx

xxx

xϕ


4

2) We solved the Poisson equation for the streamfunction with a boundary condition that assured cancellation of the normal component of the velocity field on the wall (�=const, e.g �=0 and � =Q, where Q is a flow rate ).

The velocity at the grid nodes is calculated by central difference:

3) The value of the velocity from the grid nodes is interpolated to the position of the particles:

where lh is the base function of the lagrangian interpolation. In the present work we took as lh(x) the B-spline �(x) of order one. This ends the computation process for the VIC algorithm. To satisfy the non-slip condition on the wall (��/�n=0) we utilised the vorticity generation process. In numerical practice when fluid equations are formulated in �-� terms, one comes to the problem of the determination of the vorticity value on the wall. In literature we can find a whole family of different approximate formulas allowing one to make this [13, 23]. One of the oldest and simplest is the Thom`s formula [23]:

Formula (17) may by obtained from equation (14) when one writes it on the wall and takes into account that (�i,1-�i,-1)/(2�y2)=0 (index -1 is related to the “ghost” point outside the computational domain). At each time step it is assumed that the formula designates the proper amount of vorticity on the wall. If old vortex particles that already exist in the flow give to the boundary point (by redistribution) the vorticity �old then to the nodes point on the boundary is added the new portion of vorticity:

The new portion of vorticity �new is redistributed among the nv vortex particles giving them the circulation �=(� new h2)/nv where nv was chosen in such a way that |�|<0.05, (nv=2-11). A similar process generation of vorticity was successfully applied in papers [5, 27] . Instead of formula (17) the Woods formula was also tested: �B = -3�1/�y2-(1/2)�1 [13, 23] where �1, �1 correspond to the stream function and vorticity values at distance �y from the wall. Woods formula has the same order of accuracy and gave results similar to the Thom’s formula (17).

The final step obtaining of the solutions is the displacement of the vortex particles in accordance with formula (6) and the whole process begin again from step 1). For the solution of the Poisson equation fast elliptic solver was used. To be able to solve the Poisson equation in irregular region capacitance matrix technique was used. The capacitance matrix technique is well described in many place in literature [3 ,24, 27].

)14(ωψ −=∆

)16()()( 2

jphnj

J j p

n xxlu = x u −�∈

)17(2

1,ψω i2B y

∆

−=

)18(ωωω oldnewB + =

)15(2

),(),(2

),(),(

2121

2121

h

yhxyhxv

h

hyxhyxu

jjjjj

jjjjj

−−+=

−−+−=

ψψ

ψψ


5

3. EXEMPLARY NUMERICAL RESULTS 3.1 Flow in channel with sudden expansion At first we will present the results related to flow over the plane symmetric expansion

(Fig.1). Fig.1 Skech of the gemetry for the flow over a plane symmetric sudden expansion.

The length of computational domain was taken as 32h where h=1 is the height of the step. The expansion ratio was W0/W=1/3 or 1/2. From the literature it is know that as the Reynolds number is increased the flow undergoes several changes [4, 12,14]. For a small Reynolds number, (Re�56) the lengths of separation regions behind each step are equal and velocity profiles are symmetrical. The growth of the Reynolds number (Re�125)

causes the loss of symmetry. One of the recalculation zones becomes larger. Further growth of the Reynolds number (Re �252) causes changes in the picture of the flow that are explained in the term of a bifurcation theory. [14] (fig. 2).

Fig.2. Averaged streamlines with velocity profiles at Re=56, Re=125 and Re=252, W=3h, W0=h. Dashed lines mean the streamlines have values less the zero and greater than 1.

y

xh

WW 0U


6

Fig.3. Comparison of the numerical results of velocity distribution across the vertical section with measurements taken

from [12], Re=56, W0/W=1/3. Calculated velocity distributions across a vertical cross section of the channel for different values of x for Re=56 were compared with experimental measurements published at paper [12] and it is presented in (fig.3). The agreement is good but one may notice that the distribution of the velocity near the wall is not as smooth as we had expected it to be. The vortex method provides a natural possibility for visualization of the flow and its analysis in terms of vorticity distribution by tracing the position of the vortex particles. Fig. 4 presents the sequence of the vortex particle positions at Re=1000, W0/W=1/2. For T=40 we can see very clearly the vortex structures that are in good qualitative agreement with the results presented in work [4] (see fig.5 below). We used the red color (dark and light) for marking the vortex particles of the negative sign, and the blue color (dark and light) for the positive sign of vortex particles. The darker points correspond to the value of circulation that is greater than the mean value, calculated separately for positive and negative vortex particles. We also carried out the calculation for flow at very a high Reynolds number (Re=105). We must say that we are conscious of the objections, which correspond to the problem of numerical diffusion, the resolution and so on. The aim of these numerical experiments was just to check the possibility of the VIC method for modeling of such a flow. As it was pointed out by Chorin [ 8], one should keep in mind the difference between modelling with vortices and numerical approximations of solution of a fluid motion equation by the vortex method. It seems that the last experiment belongs to the “modelling”. This means we tried with the help of a moderate number of vortices to get qualitative understanding of the vorticity field dynamic at a very high Reynolds number. The sequence of the vortex particles for the flow at Re=105 is presented in Fig.6. It is easy to notice presence of vorticity filaments -- thread-like structures

-1 -0.5 0 0.5 1(1.5h - y)/1.5h

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

u/U

max

numerical resultsmeasurem. for x = 5.75

for x = 7.5for x=8.5for x=16


7

that are regarded as typical structures of two-dimensional turbulence [2, 16, 20, 22 ,21]. One can observe that filaments are accompanied by a large coherent vortex structure that stabilise them , [16, 20, 22]. These large vortex structures are build with the both sings of vortex particles. Vortices of the same sign may undergo merging and vortices of opposite sign may form dipoles [20, 22].

Fig. 4 Evolution of the vorticity in channel with sudden symmetric expansion, Re=1000, W0/W=1/2. N means the number of particles.

Fig.5 Scanned picture of the experimental visualization from the Cherdron, Drust, Withelow paper [4](figure.9d).

All these mentioned phenomena we may find in the description of two-dimensional turbulence in literature [2, 16, 20, 22]. Fig.6 it was presents the sequence of the vortex particle positions at Re=105 for the flow over the backward step. The height of the step was h=0.5, the length of the channel was 24h, �x=�y=0.05,�t=0.01. In inlet the velocity was U=1,(Re=Uh/�). At the first frame for T=5 we can see the separation of the vortex sheet from the sharp corner and its roll-up. At the same time on the opposite wall, one can notice the concentration of the vorticity, which grows with time, and on the next frame we see that it is the place from where the process of vortex shedding goes on. One notices that the vorticity has a tendency to create filaments. The filaments of the vorticity are regarded as the fundamental structure of two-dimensional turbulence [2, 16, 20, 22]. We see that filaments are accompanied by large coherent vortex structures that stabilize them [16]. These large vortex structures


8

Fig.6 The sequence of the vortex particle position at Re=100000, W0/W=0.5

are built with the both sings, positive and negative, of vortex particles. In computer animation film one may notice that vortices of the same sign undergo merging, and vortices of the opposite sign form a dipole structure [16, 20]. All these phenomena one can find in the description of two-dimensional turbulence [2, 20, 22].

3.2 Flow over the backward-facing step For the flow over the backward-facing step there are well-documented experimental [1, 11] and numerical data [15, 21] available from literature. So this flow is a good example for testing the program. At first we checked the lengths of the recirculation zone behind the step at different Reynolds number. It is known that the reattachment length xr = xl/Hs (see fig.7) increases approximatly linearly as the Reynolds number increases.


9

Figure 7. Sketch of the geometry for the flow over backward-facing step At Fig.8 we showed the averaged streamlines for different Reynolds numbers (Re=100,200,300,400,500,600). For calculation purposes we took �t=0.01, �x=�y=0.05, L=12H, H=1,U=1. To reduce the statistical perturbations due to stochastic manner of solution of the diffusion equation we carried out an averaging process. The averaging was done for 200 time steps form T=38-40. The linear growth of the recirculation zone It is visible. For Reynolds number Re <300 the agreement with the experiments are very good [1,11] but for Re>300 the length of the recirculation zone is underestimated.

Fig. 8 Averaged streamlines at Re=100,200,300,400,500,600, t�[38,40], �t=0.01.

L

xl

U


10

It is difficult to indicate one special reason for that. It is known that for Reynolds number Re�229 [11] the velocity near the re-attachment point starts to oscillate. Due to vorticity generation on the wall and the stochastic manner of simulation of viscosity of the fluid, the flow is permanently perturbed and it is difficult precisely determine the position of the re-attachment point. The same effect was observed, in the direct vortex method used for the simulation of this flow [18]. In order to see the qualitative changes that the vorticity field undergoes when the Reynolds number increased we present in fig, 9 the vorticity field created by the vortex particle position at different Reynolds numbers in the same time T=40. In the first frame (Re=100), the vortex particles are uniformly spread throughout the channel. When the Reynolds number increases, we see that a potential core (the space without any vortex particles) appears. This potential core grows when the Reynolds number is increased. It is easy to notice the filaments of vorticity and the large vortex structure at high Reynolds numbers.

Fig. 9 The vorticity field created by vortex particle positions at different Reynolds number and in the same

dimensionless time T=40.

In Fig.10 the sequence of the instantaneous vortex particle position was shown at Reynolds number Re=105. It is interesting to notice the vorticity shedding phenomena from the point on the upper wall (opposite to the corner of the step). The remarks that I made at the end of section 3.1 about the behaviuor of the vorticity at large Reynolds numbers is also true in this case. The vorticity has a tendency to create filaments. We see that filaments are accompanied by large coherent vortex structures that stabilize them, and vortices of the opposite sign form a dipole structure [20,22, 16]. These large vortex structures are built with both positive and negative signs of vortex particles.


11

Fig.10 The sequence of the vortex particle positions in channel over backward-facing step, Re=105. 3.2 Flow over a system of buildings In paper [29] interesting pictures of the flow over the system of buildings were published. The flow was visualized using smoke (see fig.11 for scanned pictures from that paper). The pictures illustrate how minor design modifications can make a large difference in wind velocity at the pedestrian level (between the building). In the paper [ ] was notices that “the high buildings causes the high velocities by deflecting the upper and faster atmospheric layers down to the ground; where they impinge on the ground, the velocities may be double the value they would be in the absence of the building”.


12

Fig.12 shows the scheme of the configuration of the buildings that was taken to the calculation. As a unit length the height h of the smaller building was taken, h=1. The height of the higher building was 3h. On the inlet it was taken the velocity U=1, Re=Uh/�. The dimension of the computational domain was taken 15h x 6h.On the upper boundary it was taken that the normal velocity is zero. So the generation of the vorticity took place only at the rigid boundary in the bottom and on the serface of the building. Gird steps was taken as �x=�y=0.1, �t=0.01.

Fig.11. Scanned pictures of the flow visualization around two models of a tall buildings taken from the paper by

H.Thomann [29].

Fig12. The scheme of the computational domain for the flow over the system of buildings a) the plain system of

buildings b) modified system of buildings.

Fig. 13 it shows the streamlines that were obtained by vortex method. The shapes of the streamlines and generated structures are in good qualitative agreement with the experimental pictures presented in Fig.11. For numerical calculation we choose Re=2000, Re=Uh/�. The vortex method has the natural possibility of analyzing the flow features in term of vorticity through vortex particle positions. In fig.14 and 15 the sequence of the vortex particle positions for the plain (unmodified) and modified system of building was presented. We started the calculation from the

X

Y

0 5 10 150

1

2

3

4

5

6

U

a)

X

Y

0 5 10 150

1

2

3

4

5

6

Ub)


13

Fig.13. Streamlines of the flow over the buildings obtained from the calculation. To better visualize the structures of streamlines the zebra technique of drawing was used.

potential flow. For t>0 the viscosity of the fluid started to play a role. One can see the development of the Kelvin-Helmholtz type vortex structure. This vortex structure induced an air velocity greater than the velocity of the air coming toward the fronts of the buildings. This is clearly visible in Fig.16 where the close-up of the frame for T=20, together with the velocity field, was presented. A small element in front of the tall building was added to lower the velocity on the pedestrian level between the buildings. That velocity was reduced several times. It is interesting to notice the influence of that small construction on the vorticity distribution behind the tall building (compare the relevent frames in fig.14 and 15). We notice that the added construction destabilized the vortex Kelvin-Helmholtz structure, and the distribution of the vorticity behind the tall building is more chaotic.


14

Fig. 14 The sequence of the instantaneous position of the vortex particles.), Re=10000.


15

Fig. 15 The sequence of the instantaneous position of the vortex particles for modified structure of buildings, Re=10000.

Fig.16 Instantaneous vorticity field that was created by the vortex particle positions and related to its the velocity field.

4. CONCLUDING REMARKS

It seems that now it is not far from the creation of a general flow simulation package based on the vortex

method into which the user need only enter minimal data concerning boundaries in order to be able to perform the

numerical investigations. The present paper is a move in that direction. Vortex methods provide natural, useful

tools for analysing flow in term of vorticity dynamics, and the visualisation of the flow by vortex particles. The

study of the evolution of the vorticity field helps one to understand the features of flow in a complicated geometry.

It is one of the few methods which give reasonable results at large interval of Reynolds numbers trying to solve the

Navier-Stokes directly. Further study on the VIC method should clarify the problem of numerical diffusion that


16

may be introduced by a numerical grid. It is believed that introduction of the deterministic method of the diffusion

simulation instead of stochastic one should improve the numerical results.

REFERENCES [1] Armaly B.F., Durst f., Pererira J.C., Schönung B., Experimental and Theoretical Investigations of

Backward-Facing Step Flow, J. Fluid Mech., vol. 127, pp.473-496, (1983). [2] Brachet M.E., Meneguzzi M., Politano H., Sulem P..L., The dynamics of freely decaying two-dimensional

turbulence, vol. 194, pp. 333-349, (1988). [3] Buzbee B. L., Dorr F.W.,George J. A., Golub G. H.: The solution of the discrete Poisson equation on

irregular regions, SIAM, J. Num. Anal., vol. 8, pp. 722-736, (1971) [4] Cherdron W., Durst F., Whitelaw J.H.: Asymmetric flows and instabilities in symmetric ducts with sudden

expansions, J.Fluid Mech., vol. 84, pp. 13-31,(1978). [5] Chang C. C., Chern R.-L.: A numerical study of flow around an impulsively started circular cylinder by a

deterministic vortex method, J. Fluid Mech., vol. 233, pp. 243-263, (1994). [6] Chorin A. J. Numerical Study of Slightly Viscous Flow, J. Fluid Mech. Vol. 57, pp. 785-796, (1973). [7] Chorin A.J., Marsden J.E.: A Mathematical Introduction to Fluid Mechanics, Springer (1979). [8] Chorin A. J., Microstructure, renormalization, and more efficient vortex methods, ESAIM: Proceedings,

Vortex Flows and Related Numerical Methods II, 96, Vol. 1, pp. 1-14, (1996), http:// www.emath.fr/proc/ Vol.1/contents.htm

[9] Christiansen J.P.: Numerical simulation of Hydrodynamics by the Method of Point Vortices, J.1 Comp. Phys., vol. 13, pp. 363-379, (1973).

[10] Cottet G-H. A A particle-grid superposition method for the Navier-Stokes equations, J. Comp. Phys., vol. 89, pp. 301-318, (1980).

[11] Denham H.K., Patrick M.A., Laminar flow over a downstream-facing step in a Two-dimensional Flow Channel, Trans. Instn Chem. Engrs., vol.52, pp. 361-367,(1974).

[12] Durst F., Melling A., Whitelaw J.H.: Low Reynolds number flow over a plane symmetric sudden expansion, J,Fluid Mech,, vol. 64, pp. 111-128, (1974).

[13] E Weinan, Liu Jian-Guo: Vorticity boundary condition and related issues for finite difference schemes, J. Comput. Phys., vol 124, s. 368-382, (1996)

[14] Faren R.M., Mullin T., Cliffe K.A.: Nonlinear flow phenomena in a symmetric sudden expansion, J.Fluid Mech., vol. 124, s. 368-382, (1990)

[15] Ghoniem A.F., Cagnon V., Vortex Simulations of Laminar Recirculating Flow, J. Comp. Phys., vol.68, pp. 348-377, (1987).

[16] Kevlahan N. K.-R. Farge M.: Vorticity filaments in two-dimensional turbulence: creation, stability and effect, J. Fluid Mech., vol. 346, pp. 49-76, (1997)

[17] Kloeden P., Platen E.: Numerical Solution of Stochastic Differential Equations, Springer, 1982 [18] Kong J. X., Contribution a l'analyse numericque des methodes de couplage particules-grille en mecanique

des fluides, Phd. dissert., Universite J. Fourier Grenoble, LMC-IMAG (1993). [19] Kudela H., Modelowanie zjawisk hydrodynamicznych metodami dyskretnych wirów, Oficyna Wydawicza

Politechniki Wroc»awskiej , seria: monografie 25, (1995). [20] Lesieur M., Turbulence in Fluids, Third edition, Kluwer Academic Publishers, (1997) [21] Kaiktsis L., Karniadakis G.E.,Orszag S. A., Onset of three-dimensionality, equilibria, and early transition in

flow over a backward-facing step, J. Fluid Mech. Vol. 231, pp. 501-528, (1991) [22] McWilliams J.C.:The emergence of isolated coherent vortices in turbulent flow, J. Fluid Mech., vol. 146,

pp.21-46, (1984). [23] Peyret R., Taylor T. D.: Computational methods for fluid flow, Springer-Verlag (1983) [24] Proskurowski W., Widlund O.: On the numerical solution of Helmholtz’s equation by the capacitance


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matrix method, Math. Comput., vol.30, pp.433-468, (1976) [25] Raviart P.A.: An analysis of particle methods, in Numerical methods in fluid Dynamics , ed. Brezzi F.,

Lecture Notes in Mathematics, vol. 1127, pp. 243-324, Spinger Verlag, Berlin, (1985) [26] Schoenberg I.J.: Contributions to the problem of approximation of equidistant data by analytic function,

Part A, Quart. Appl. Math., vol. 4. pp. 45-99, (1946). [27] Schumann U., Sweet R.A.: Direct Poisson Equation Solver for Potential and Pressure Fields on a

Staggered Grid with Obstacles, Lecture Notes in Physics, vol.59, s. 397-403, (1976). [28] Smith P.A., Stansby P.K.: Impulsively started flow around a circular cylinder by the vortex method, J. Fluid

Mech. , vol. 194, pp.45-77, (1998) [29] Thomann H., Wind effects on buildings and structures, American Scientist, vol.63, s. 278-287, (1975)

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